H. Mckean, A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS, Proceedings of the National Academy of Sciences, vol.56, issue.6, pp.1907-1911, 1966.
DOI : 10.1073/pnas.56.6.1907

H. Mckean, Propagation of chaos for a class of non-linear parabolic equationsInStochastic Differential Equations, [Lecture Series in Differential Equations, pp.41-57

W. Braun and K. Hepp, The Vlasov dynamics and its fluctuations in the 1/N limit of interacting classical particles, Communications in Mathematical Physics, vol.35, issue.2, pp.101-113, 1977.
DOI : 10.1007/BF01611497

D. Dawson, Critical dynamics and fluctuations for a mean-field model of cooperative behavior, Journal of Statistical Physics, vol.11, issue.1, pp.29-85, 1983.
DOI : 10.1007/BF01010922

R. Dobrushin, Prescribing a system of random variables by conditional distributions. Theory Probab Appl, pp.458-486, 1970.

H. Tanaka, Probabilistic treatment of the Boltzmann equation of Maxwellian molecules. Probab Theory Relat Fields, pp.67-105, 1978.

H. Tanaka and M. Hitsuda, Central limit theorem for a simple diffusion model of interacting particles, HiroshimaMathJ1981, vol.11, issue.2, pp.415-423

H. Tanaka, Some probabilistic problems in the spatially homogeneous Boltzmann equation.I n Theory and Application of Random Fields Lecture Notes in Control and Information Sciences, pp.258-267

H. Tanaka, Limit theorems for certain diffusion processes with interaction. Stochastic Analysis. Amsterdam: North-Holland, pp.469-488

A. Sznitman, Nonlinear reflecting diffusion process, and the propagation of chaos and fluctuations associated, Journal of Functional Analysis, vol.56, issue.3, pp.311-336, 1984.
DOI : 10.1016/0022-1236(84)90080-6

A. Sznitman, A propagation of chaos result for Burgers' equation. Probab Theory Relat Fields, pp.581-613, 1986.

A. Sznitman, E. Pardoux, A. Sznitman, and . Berlin, Topics in propagation of chaos.I nEcole d'Eté de Probabilités de Saint-Flour XIX 1989, Lecture Notes in Math, vol.1991, issue.1464, pp.165-251

A. Ecker, P. Berens, G. Keliris, M. Bethge, N. Logothetis et al., Decorrelated Neuronal Firing in Cortical Microcircuits, Science, vol.327, issue.5965, p.584, 2010.
DOI : 10.1126/science.1179867

A. Hodgkin and A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, The Journal of Physiology, vol.117, issue.4, pp.500-544, 1952.
DOI : 10.1113/jphysiol.1952.sp004764

R. Fitzhugh, Theoretical Effect of Temperature on Threshold in the Hodgkin-Huxley Nerve Model, The Journal of General Physiology, vol.49, issue.5, pp.989-1005, 1966.
DOI : 10.1085/jgp.49.5.989

R. Fitzhugh, Mathematical models of excitation and propagation in nerve.I nBiological Engineering . Edited by Schwan HP, pp.1-85

E. Izhikevich, Dynamical Systems in Neuroscience: The Geometry of Excitability and Bursting, 2007.

L. Lapicque, Recherches quantitatifs sur l'excitation des nerfs traitee comme une polarisation, J. Physiol. Paris, vol.9, pp.620-635, 1907.

H. Tuckwell, Introduction to Theoretical Neurobiology, 1988.

R. Fitzhugh, Mathematical models of threshold phenomena in the nerve membrane, The Bulletin of Mathematical Biophysics, vol.124, issue.4, pp.257-278, 1955.
DOI : 10.1007/BF02477753

J. Nagumo, S. Arimoto, and S. Yoshizawa, An Active Pulse Transmission Line Simulating Nerve Axon, Proceedings of the IRE, vol.50, issue.10, pp.2061-2070, 1962.
DOI : 10.1109/JRPROC.1962.288235

A. Destexhe, Z. Mainen, and T. Sejnowski, Synthesis of models for excitable membranes, synaptic transmission and neuromodulation using a common kinetic formalism, Journal of Computational Neuroscience, vol.87, issue.3, pp.195-230, 1994.
DOI : 10.1007/BF00961734

E. Kandel, J. Schwartz, and T. Jessel, Principles of Neural Science, 2000.

J. Cox, J. Ingersoll, . Jr, and S. Ross, A Theory of the Term Structure of Interest Rates, Econometrica, vol.53, issue.2, pp.385-407, 1985.
DOI : 10.2307/1911242

X. Mao, Stochastic Differential Equations and Applications Chichester: Horwood, 2008.

S. Amari, Characteristics of random nets of analog neuron-like elements, IEEE Trans Syst Man Cybern, vol.2, issue.5, pp.643-657, 1972.

S. Amari, Dynamics of pattern formation in lateral-inhibition type neural fields, Biological Cybernetics, vol.13, issue.2, pp.77-87, 1977.
DOI : 10.1007/BF00337259

H. Wilson and J. Cowan, Excitatory and Inhibitory Interactions in Localized Populations of Model Neurons, Biophysical Journal, vol.12, issue.1, pp.1-24, 1972.
DOI : 10.1016/S0006-3495(72)86068-5

H. Wilson and J. Cowan, A mathematical theory of the functional dynamics of cortical and thalamic nervous tissue, Kybernetik, vol.12, issue.2, pp.55-80, 1973.
DOI : 10.1007/BF00288786

A. Hammerstein and . Integralgleichungen-nebst-anwendungen, Nichtlineare Integralgleichungen nebst Anwendungen, Acta Mathematica, vol.54, issue.0, pp.117-176, 1930.
DOI : 10.1007/BF02547519

O. Faugeras, F. Grimbert, and J. Slotine, Absolute Stability and Complete Synchronization in a Class of Neural Fields Models, SIAM Journal on Applied Mathematics, vol.69, issue.1, pp.205-250, 2008.
DOI : 10.1137/070694077

URL : https://hal.archives-ouvertes.fr/inria-00423423

S. Coombes and M. Owen, Bumps, Breathers, and Waves in a Neural Network with Spike Frequency Adaptation, Physical Review Letters, vol.94, issue.14, p.94148102, 2005.
DOI : 10.1103/PhysRevLett.94.148102

B. Ermentrout, Neural networks as spatio-temporal pattern-forming systems, Reports on Progress in Physics, vol.61, issue.4, pp.353-430
DOI : 10.1088/0034-4885/61/4/002

G. Ermentrout and J. Cowan, Temporal oscillations in neuronal nets, Journal of Mathematical Biology, vol.13, issue.3, pp.265-280
DOI : 10.1007/BF00275728

C. Laing, W. Troy, B. Gutkin, and G. Ermentrout, Multiple Bumps in a Neuronal Model of Working Memory, SIAM Journal on Applied Mathematics, vol.63, issue.1, pp.62-97, 2002.
DOI : 10.1137/S0036139901389495

P. Chossat and O. Faugeras, Hyperbolic Planforms in Relation to Visual Edges and Textures Perception, PLoS Computational Biology, vol.33, issue.12, p.1000625, 2009.
DOI : 10.1371/journal.pcbi.1000625.s006

URL : https://hal.archives-ouvertes.fr/hal-00807344

R. Veltz and O. Faugeras, Local/Global Analysis of the Stationary Solutions of Some Neural Field Equations, SIAM Journal on Applied Dynamical Systems, vol.9, issue.3, pp.954-998
DOI : 10.1137/090773611

URL : https://hal.archives-ouvertes.fr/hal-00712201

L. Abbott, V. Vreeswijk, and C. , Asynchronous states in networks of pulse-coupled neuron, Phys Rev, vol.48, pp.1483-1490, 1993.

D. Amit and N. Brunel, Model of global spontaneous activity and local structured activity during delay periods in the cerebral cortex, Cerebral Cortex, vol.7, issue.3, pp.237-252, 1997.
DOI : 10.1093/cercor/7.3.237

N. Brunel and V. Hakim, Fast Global Oscillations in Networks of Integrate-and-Fire Neurons with Low Firing Rates, Neural Computation, vol.15, issue.7, pp.1621-1671, 1999.
DOI : 10.1038/373612a0

N. Brunel, Dynamics of sparsely connected networks of excitatory and inhibitory spiking neurons, Journal of Computational Neuroscience, vol.8, issue.3, pp.183-208, 2000.
DOI : 10.1023/A:1008925309027

E. Boustani, S. Destexhe, and A. , A Master Equation Formalism for Macroscopic Modeling of Asynchronous Irregular Activity States, Neural Computation, vol.0, issue.0, pp.46-100, 2009.
DOI : 10.1126/science.1127241

URL : https://hal.archives-ouvertes.fr/hal-00377137

M. Mattia, D. Giudice, and P. , Population dynamics of interacting spiking neurons, Physical Review E, vol.66, issue.5, p.51917, 2002.
DOI : 10.1103/PhysRevE.66.051917

D. Cai, L. Tao, M. Shelley, and D. Mclaughlin, An effective kinetic representation of fluctuation-driven neuronal networks with application to simple and complex cells in visual cortex, Proceedings of the National Academy of Sciences, vol.101, issue.20, pp.7757-7762, 2004.
DOI : 10.1073/pnas.0401906101

C. Ly and D. Tranchina, Critical Analysis of Dimension Reduction by a Moment Closure Method in a Population Density Approach to Neural Network Modeling, Neural Computation, vol.20, issue.8, pp.2032-2092, 2007.
DOI : 10.1016/S0006-3495(72)86068-5

E. Rolls and G. Deco, The Noisy Brain: Stochastic Dynamics as a Principle of Brain Function, 2010.

W. Softky and C. Koch, The highly irregular firing of cortical cells is inconsistent with temporal integration of random EPSPs, pp.334-350, 1993.

N. Brunel and P. Latham, Firing Rate of the Noisy Quadratic Integrate-and-Fire Neuron, Neural Computation, vol.13, issue.10, pp.2281-2306, 2003.
DOI : 10.1088/0954-898X/4/3/002

H. Plesser, Aspects of signal processing in noisy neurons, 1999.

J. Touboul and O. Faugeras, A characterization of the first hitting time of double integral processes to curved boundaries, Advances in Applied Probability, vol.27, issue.02, pp.501-528, 2008.
DOI : 10.2307/3215174

J. Beggs and D. Plenz, Neuronal Avalanches Are Diverse and Precise Activity Patterns That Are Stable for Many Hours in Cortical Slice Cultures, Journal of Neuroscience, vol.24, issue.22, pp.245216-5229
DOI : 10.1523/JNEUROSCI.0540-04.2004

M. Benayoun, J. Cowan, W. Van-drongelen, and E. Wallace, Avalanches in a Stochastic Model of Spiking Neurons, PLoS Computational Biology, vol.26, issue.7, p.1000846, 2010.
DOI : 10.1371/journal.pcbi.1000846.s002

A. Levina, J. Herrmann, and T. Geisel, Phase Transitions towards Criticality in a Neural System with Adaptive Interactions, Physical Review Letters, vol.102, issue.11, p.118110, 2009.
DOI : 10.1103/PhysRevLett.102.118110

J. Touboul and A. Destexhe, Can Power-Law Scaling and Neuronal Avalanches Arise from Stochastic Dynamics?, PLoS ONE, vol.5, issue.2, p.8982
DOI : 10.1371/journal.pone.0008982.t005

URL : https://hal.archives-ouvertes.fr/hal-00466697

P. Bressloff, Stochastic Neural Field Theory and the System-Size Expansion, SIAM Journal on Applied Mathematics, vol.70, issue.5, pp.1488-1521, 2009.
DOI : 10.1137/090756971

M. Buice and J. Cowan, Field-theoretic approach to fluctuation effects in neural networks, Physical Review E, vol.75, issue.5, p.51919, 2007.
DOI : 10.1103/PhysRevE.75.051919

M. Buice, J. Cowan, and C. Chow, Systematic Fluctuation Expansion for Neural Network Activity Equations, Neural Computation, vol.13, issue.1, pp.377-426, 2010.
DOI : 10.1093/acprof:oso/9780198509233.001.0001

T. Ohira and J. Cowan, Master-equation approach to stochastic neurodynamics, Physical Review E, vol.48, issue.3, pp.2259-2266, 1993.
DOI : 10.1103/PhysRevE.48.2259

W. Gerstner, Time structure of the activity in neural network models, Physical Review E, vol.51, issue.1, pp.738-758, 1995.
DOI : 10.1103/PhysRevE.51.738

O. Faugeras, J. Touboul, and B. Cessac, A constructive mean-field analysis of multi population neural networks with random synaptic weights and stochastic inputs, Frontiers in Computational Neuroscience, vol.3, 2009.
DOI : 10.3389/neuro.10.001.2009

URL : https://hal.archives-ouvertes.fr/inria-00258345

A. Guionnet, Averaged and quenched propagation of chaos for spin glass dynamics. Probab Theory Relat Fields, pp.183-215, 1997.

A. Chizhov and L. Graham, Population model of hippocampal pyramidal neurons, linking a refractory density approach to conductance-based neurons, Physical Review E, vol.75, issue.1, p.11924, 2007.
DOI : 10.1103/PhysRevE.75.011924

URL : https://hal.archives-ouvertes.fr/hal-00174130

H. Sompolinsky, A. Crisanti, and H. Sommers, Chaos in Random Neural Networks, Physical Review Letters, vol.61, issue.3, pp.259-262, 1988.
DOI : 10.1103/PhysRevLett.61.259

H. Sompolinsky and A. Zippelius, Relaxational dynamics of the Edwards-Anderson model and the mean-field theory of spin-glasses, Phys Rev B Condens Matter Mater Phys, issue.11, pp.256860-6875, 1982.

F. Bolley, J. Cañizo, and J. Carrillo, STOCHASTIC MEAN-FIELD LIMIT: NON-LIPSCHITZ FORCES AND SWARMING, Mathematical Models and Methods in Applied Sciences, vol.21, issue.11, pp.2179-2210, 2011.
DOI : 10.1142/S0218202511005702

D. Talay and O. Vaillant, A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations, The Annals of Applied Probability, vol.13, issue.1, pp.140-180, 2003.
DOI : 10.1214/aoap/1042765665

URL : https://hal.archives-ouvertes.fr/inria-00072260

M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Mathematics of Computation, vol.66, issue.217, pp.157-192, 1997.
DOI : 10.1090/S0025-5718-97-00776-X

M. Hutzenthaler, A. Jentzen, and P. Kloeden, Strong and weak divergence in finite time of Euler's method for stochastic differential equations with non-globally Lipschitz continuous coefficients, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, vol.316, issue.3, pp.4671563-1576, 2011.
DOI : 10.1093/imanum/drl032

M. Hutzenthaler and A. Jentzen, Convergence of the Stochastic Euler Scheme for Locally Lipschitz Coefficients, Foundations of Computational Mathematics, vol.30, issue.3, pp.657-706, 2011.
DOI : 10.1007/s10208-011-9101-9

W. Schiesser, The Numerical Method of Lines: Integration of Partial Differential Equations.S a n Diego, 1991.

W. Schiesser and G. Griffiths, A Compendium of Partial Differential Equation Models: Method of Lines Analysis with Matlab, 2009.
DOI : 10.1017/CBO9780511576270

C. Ueberhuber, Numerical Computation 2: Methods, Software, and Analysis, 1997.
DOI : 10.1007/978-3-642-59109-9

K. Morton and D. Mayers, Numerical Solution of Partial Differential Equations: An Introduction, 2005.
DOI : 10.1017/CBO9780511812248

J. Touboul, G. Hermann, and O. Faugeras, Noise-Induced Behaviors in Neural Mean Field Dynamics, SIAM Journal on Applied Dynamical Systems, vol.11, issue.1, pp.49-81
DOI : 10.1137/110832392

URL : https://hal.archives-ouvertes.fr/hal-00846146

J. Baladron, D. Fasoli, and O. Faugeras, Three applications of GPU computing in neuroscience, Comput Sci Eng, vol.14, pp.40-47, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00845582

S. Herrmann and J. Tugaut, Non-uniqueness of stationary measures for self-stabilizing processes. Stoch Process Appl, pp.1215-1246, 2010.

K. Pakdaman, M. Thieullen, and G. Wainrib, Fluid limit theorems for stochastic hybrid systems with application to neuron models, Advances in Applied Probability, vol.46, issue.03, pp.761-794, 2010.
DOI : 10.1073/pnas.0236032100

URL : https://hal.archives-ouvertes.fr/hal-00447808

G. Wainrib, Randomness in neurons: a multiscale probabilistic analysis Ecole Polytechnique, 2010.

J. Goldwyn, N. Imennov, M. Famulare, and E. Shea-brown, Stochastic differential equation models for ion channel noise in Hodgkin-Huxley neurons, Physical Review E, vol.83, issue.4, p.41908, 2011.
DOI : 10.1103/PhysRevE.83.041908

L. Evans, Partial Differential Equations, Graduate Studies in Mathematics, vol.19, 1998.